Example 2.3 (Limit groups of Sela) Limit groups are a particular class of finitely presented groups that arise in the study of first order logical theory of finitely generated groups - they can be defined as those groups that share the same theory as a free group [1]}. It is known that such groups are precisely those groups \(G\)  that admit an asymptotically faithful covering family \(\lbrace \pi _{m}:G \rightarrow G_{m}\rbrace _{m}\)  where each \(G_{m}\)  is a finitely generated free group (of a fixed rank) - a condition in this case that appears with the terminology “fully residually free” in the literature. However, using the generators of \(G\)  to generate each \(G_{m}\)  is necessary for the asymptotically faithful covering example, and this will not be a free generating set.
